Volume 2010, Article ID 289571,13pages doi:10.1155/2010/289571
Research Article
Time-Frequency Data Reduction for Event Related Potentials:
Combining Principal Component Analysis and Matching Pursuit
Selin Aviyente,
1Edward M. Bernat,
2Stephen M. Malone,
3and William G. Iacono
31Department of Electrical and Computer Engineering, Michigan State University East Lansing, MI 48824, USA 2Department of Psychology, Florida State University, Tallahassee, FL 32306, USA
3Department of Psychology, University of Minnesota, Minneapolis, MN 55455, USA
Correspondence should be addressed to Selin Aviyente,[email protected]
Received 2 February 2010; Revised 30 March 2010; Accepted 5 May 2010
Academic Editor: Syed Ismail Shah
Copyright © 2010 Selin Aviyente et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Joint time-frequency representations offer a rich representation of event related potentials (ERPs) that cannot be obtained through
individual time or frequency domain analysis. This representation, however, comes at the expense of increased data volume and
the difficulty of interpreting the resulting representations. Therefore, methods that can reduce the large amount of time-frequency
data to experimentally relevant components are essential. In this paper, we present a method that reduces the large volume of ERP time-frequency data into a few significant time-frequency parameters. The proposed method is based on applying the widely used matching pursuit (MP) approach, with a Gabor dictionary, to principal components extracted from the time-frequency domain. The proposed PCA-Gabor decomposition is compared with other time-frequency data reduction methods such as the time-frequency PCA approach alone and standard matching pursuit methods using a Gabor dictionary for both simulated and biological data. The results show that the proposed PCA-Gabor approach performs better than either the PCA alone or the standard MP data reduction methods, by using the smallest amount of ERP data variance to produce the strongest statistical separation between experimental conditions.
1. Introduction
Event-related potential (ERP) signals measured at the scalp are produced by partial synchronization of neuronal field potentials across the cortex [1]. This synchronization medi-ates the “top-down” and “bottom-up” communication both within and between brain areas and has particular impor-tance during the anticipation of and attention to stimuli or events. Event related potentials (ERPs) are obtained by averaging EEG signals recorded over multiple trials or epochs time-locked to the particular stimulus. ERP signal analysis has proven to be effective in assessing the brain’s current functional state and reflect many pathological processes (e.g., [2–6]).
Typically, ERP analysis is performed in the time domain, where the amplitudes and latencies of prominent peaks in the averaged potentials are usually measured and correlated with information processing mechanisms. However, this conventional approach has two major shortcomings. First,
it is well-known that ERPs are transient and nonstationary signals. Second, ERPs generally contain multiple overlapping processes operating across different time and frequency ranges. A primary approach to this problem has been to utilize time-frequency signal representations to detect transient activity and to disentangle overlapping processes. Several methods exist to fulfill this goal including wavelet and wavelet packet decomposition [7–11], sparse signal representations using overcomplete dictionaries (such as matching pursuit [12,13] and basis pursuit [14]), Cohen’s class of time-frequency distributions [15, 16], and the recently introduced high resolution time-frequency distribu-tions [17–20].
tradeoff, that is, at high frequencies, the temporal resolution is high whereas the frequency resolution is low and vice versa for low frequencies. Sparse representations such as matching pursuit and basis pursuit aim to find a “best” fit to the given signal in terms of the elements of a redundant family of functions, called the dictionary [12,13]. The “best” fit to the given signal is quantified through both the mean square error between the representation and the actual signal and the sparseness of the representation, that is, the number of elements of the dictionary used in the representation should be minimal. This approach has the advantage of offering a fully quantitative description of the ERPs by parameterizing the time-frequency plane at the expense of being compu-tationally expensive. Cohen’s class of distributions provides advantages over the other time-frequency representations in that it accurately characterizes the physical time-frequency properties of a signal, for example, energy and marginals, and yields uniformly high resolution over the entire time-frequency plane [15,22]. Recently, time-frequency distribu-tions with improved resolution and concentration around the instantaneous frequency have been introduced such as the reassigned time-frequency representations, higher order polynomial distributions, and complex-lag distributions [17,
20,23]. Although these methods improve the resolution of the representations, they come at the expense of increased computational complexity and in some cases losing some of the desirable properties such as the marginals. Moreover, these distributions have been shown to be the most effective for polynomial phase signals whereas ERPs have been shown to be well represented by damped sinusoids [24], thus the improvement provided by these more complex distributions would be minimal. For these reasons, in this paper we will focus on the Cohen’s class of distributions, in particular the Reduced Interference Distributions.
The high resolution provided by Cohen’s class of time-frequency distributions come at the expense of increased data. The application of these distributions to large sets of ERP data has tended to rely on a time-frequency region of interest (TF-ROI; region of interest on the TF surface) to define activity for evaluation. Therefore, there is a growing need for data reduction and feature extraction methods for reducing the three dimensional time-frequency surfaces of ERPs to a few parameters. The problem of feature extraction and data reduction has been traditionally addressed using parametric and nonparametric methods. Parametric approaches include sparse representations using overcomplete dictionaries [12–14, 25–28], extraction of features from the time-frequency distributions such as the energy in different frequency bands, computation of higher order joint moments [29, 30], and entropy [31]. Non-parametric data reduction methods, on the other hand, include data-driven multivariate component analysis such as the application of matrix factorization methods to time-frequency distributions. These methods include the non-negative matrix factorization (NMF) [32–34], singular value decomposition (SVD) [35], independent component analy-sis (ICA) [1,36], and principal component analysis (PCA) [37–40] to extract time-frequency features for classification purposes or for reducing the time-frequency surfaces to
a few meaningful components. The application of the matrix factorization approaches have been mostly limited to decom-posing a single time-frequency matrix into significant time and frequency components to reduce the dimensionality and extract features for consequent classification [34, 39]. However, in ERP analysis there is a need for multivariate processing, that is, it is important to extract components that describe a collection of signals, such as those collected over multiple channels or multiple subjects. The principal component analysis of time-frequency vectors representing multiple subjects described in [37,41] addresses this issue by extracting time-frequency principal components over a collection of ERP waveforms. At this point, it is also important to motivate the use of PCA over other data factorization methods. PCA is a multivariate technique that seeks to uncover latent variables responsible for patterns of covariation in the data set and has been used widely for time domain ERP data description and reduction [42,43]. It is commonly applied to the covariance of the data matrix and is thus similar to SVD in the extracted components. PCA does not make any strong assumptions about the data, unlike NMF which imposes nonnegativeness, with the only assumption being that the observations are linear functions of the extracted components which is a common assumption in ERP analysis. ICA has been proposed as a promising alternative to PCA for ERP data reduction [1,36]. However, recent comparisons of PCA with ICA for ERP data analysis indicates that ICA suffers from the component ”splitting” problem, that is, components that should not be separated are split into multiple components, and that it is more suitable for spatial decompositions rather than temporal ones [44,45]. Further, ICA has been most commonly applied to time-domain ERP signal representations, and its use with time-frequency ERP representations has not been well-validated. For these reasons, in the current paper we use PCA as the first step in our data reduction algorithm.
on the time-frequency plane [22,46]. Second, the parameters of the Gabor logons are well-suited for identifying between transient versus oscillatory brain activity as well as separating between overlapping time-frequency events with varying duration or frequency oscillation. They have been widely used in time-frequency representation of ERP signals [47–
49], in particular EEG phenomena including sleep spindles [13,50] and epileptic seizures [51]. An algorithm similar to matching pursuit is developed in the time-frequency plane to determine the best set of logons that describe each ERP time-frequency principal component [12]. Fitting Gabor logons to the extracted principal components offers three potential benefits. First, decomposing the principal components (PCs) into a few logons would capture the major activity described by that principal component while at the same time serve as a tool of denoising, that is, removing the unwanted noise or activity that may exist in the principal component. Second, insofar as a single logon can characterize the primary activity in the experimental manipulations for each principal component, this would offer evidence that the principal components approach is efficient at extracting compact time-frequency representations. Finally, the extracted logons offer an important unit of analysis in their own right, in that they are maximally compact by definition. The proposed methods are compared to both parametric and nonpara-metric data reduction methods in the time-frequency plane, namely, the standard matching pursuit algorithm [12, 52] and PCA in terms of efficiency, computational complexity and the effectiveness in describing the experimental effects in the data. To evaluate these methods, we employ both biological [41] and simulated data [37], that have been previously evaluated using the PCA on the time-frequency plane approach.
The rest of this paper is organized as follows.Section 2
gives a brief review of time-frequency distributions and various matching pursuit approaches. Section 3introduces the data reduction method proposed in this paper, com-bining principal component analysis with matching pursuit on the time-frequency plane. Section 4 details the data analyzed in this paper and presents the results of applying the proposed method to both simulated data and ERP signals. A comparison with different time-frequency data reduction methods is also given in this section. Finally, Section 5
concludes the paper and discusses the major contributions.
2. Background
2.1. Time-Frequency Distributions. A bilinear time-frequen-cy distribution (TFD), C(t,ω), from Cohen’s class can be expressed as (all integrals are from−∞to∞unless otherwise stated) [22]
C(t,ω)=
φ(θ,τ)x
u+τ 2
x∗
u−τ
2
×ej(θu−θt−τω)du dθ dτ,
(1)
whereφ(θ,τ) is the kernel function in the ambiguity domain (θ,τ), x(t) is the signal, and t and ω are the time and the frequency variables, respectively. Some of the most
desired properties of TFDs are the energy preservation, the marginals, and the reduced interference. For bilinear time-frequency distributions, cross-terms occur when the signal is multicomponent, that is, ifx(t) = N
i=1xi(t) then C(t,ω) = N
i=1Cxi,xi(t,ω) +
i /=j2 Re(Cxi,xj(t,ω)), where
Cxi,xi and Cxi,xj refer to the autoterms and cross-terms, respectively. The cross-terms will introduce time-frequency structures that do not correspond to the time-frequency spectrum of the actual signal. For this reason, in this paper we will use reduced interference distributions (RIDs) that concentrate the energy across the autoterms, satisfy the energy preservation and the marginals [53].
2.2. Matching Pursuit. The matching pursuit algorithm, originally proposed by Mallat, aims at obtaining the “best” linear representation of a signal in terms of functions,
{gi}i=1,2,...,N (sometimes referred to as atoms), from an
over-complete dictionary,D, using an iterative search algorithm [12].
(1) Define the 0th order residual asR0x=x,D=Dand
setk=0.
(2) For thekth order residual,Rkx, select the best atom
such that the inner product between the residual and the atom is maximized
gk=argmax
gi∈D
Rkx,gi . (2)
(3) Compute the residueRk+1xas
Rk+1x=Rkx−Rkx,gk gk. (3)
(4) Setk=k+ 1,D=D\gk, and go back to step 2 until a predetermined stopping criterion is achieved. The stopping criterion can either be a preselected number of atoms to describe the signal or a percentage of energy of the original signal described by the selected atoms. After M iterations, the following linear representation is obtained:
x= M
k=1
Rkx,gk gk+RM+1x. (4)
This procedure converges to x in the limit, that is, x =
∞
k=1Rkx,gkgk, and preserves signal energy.
2.3. Simultaneous Matching Pursuit. The principle of MP can easily be generalized to the simultaneous decomposition of multiple signals, X = (x1,x2,. . .,xr), into atoms from
method will be applied to multiple ERPs from different subjects and not from multiple channels. This algorithm can be described as follows.
(1) Define for each signal xl the 0th order residual as R0xl=xland setD=D,k=0.
(2) For thekth order residual,Rkxl, select the best atom
such that the sum of the squared inner products between the atom and the residual from each signal is maximized
gk=argmax
gi∈D
r
l=1
Rkxl,gi 2.
(5)
(3) Compute the residueRk+1xlfor each signal:
Rk+1xl=Rkxl−Rkxl,gk gk. (6)
(4) Setk=k+ 1,D=D\gk, and go back to step 2 until a predetermined stopping criterion is achieved. The stopping criterion can either be a preselected number of atoms to describe the collection of signals or an average percentage of energy of the original signals described by the selected atoms. AfterM iterations, the following linear representation is obtained for each signal:
xl= M
k=1
Rkxl,gk gk+RM+1xl. (7)
3. PCA-Gabor Method
Ideally, a time-frequency domain ERP data reduction method will faithfully reproduce established time and frequency-based findings (i.e., peaks in the time domain such as P300 or summaries of frequency activity such as alpha), and also allow a more complex view of these phenomena using the joint time-frequency information available in the TFDs. The decomposition method used in this paper is based on two stages of consecutive data reduction. The first stage is a direct extension of PCA into the joint time-frequency domain and the second stage is the parametrization of the time-frequency principal components using a matching pursuit type algorithm.
3.1. PCA on the Time-Frequency Plane. The first stage of the algorithm extends principal component analysis to the time-frequency plane as follows.
(1) Compute the time-frequency distribution of each ERP waveform from multiple subjects,xi, 1≤i≤L:
TFDi
n,ω;ψ= N
n1=−N
N
n2=−N
xi(n+n1)x∗i(n+n2)
×ψ
−n1+n2
2 ,n1−n2
e−jω(n1−n2), (8)
whereψ is the discrete-time kernel in the time and time-lag domain andxi(n) is theith ERP waveform. In this paper, the binomial kernel, given by
ψ(n,m)=2−|m| ⎛ ⎝ |m|
n+|m| 2
⎞
⎠ for|n| ≤ |m|
2 , (9)
is used as the time-frequency kernel.
(2) Given L ERP waveforms, rearrange the
time-frequency surfaces into vectors and form the matrix
X=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
TFDT1
TFDT2
.. .
TFDTL
⎤ ⎥ ⎥ ⎥ ⎥ ⎥
⎦. (10)
(3) Compute the covariance matrix,Σ=XXT.
(4) Decompose the covariance matrix using principal component analysis
Σ=
L
j=1
λjPCjPCTj, (11)
whereλj is the eigenvalue of each principal
compo-nentPCj. The principal components determine the
span of the time-frequency space.
(5) Rotate the principal components using varimax rotation [57]. Varimax rotation is an orthogonal transform that rotates the principal components such that the variance of the factors is maximized. This rotation improves the interpretability of the principal components.
(6) Rearrange each principal component into a time-frequency surface to obtain the ERP components in the time-frequency domain.
After the principal components on the time-frequency plane are extracted, they are ordered based on their eigenvalues and the most significant ones are used in the following parametrization stage. The number of principal components to keep is determined based on a normalized energy threshold.
time domain would increase the computational complexity. Second, this offers a way of directly modeling the time-frequency energy distribution.
An overcomplete dictionary of Gabor logons on the frequency plane is constructed by computing the time-frequency distribution of discrete time atomsg(n;n0,k0) =
exp(−(1/2σ2)(n −n0)2
) exp(j2π(k0/K)(n −n0)) where σ
is the scale parameter, n0 and k0 are the discrete time and frequency shift parameters, respectively, and K is the total number of frequency samples. The elements of the dictionary,D, are the binomial TFDs of these atoms,Gi(n,k). The number of elements in the dictionary are determined by the range ofn0,k0 andσ. In this paper,n0 = 1,. . .,N, whereNis the total number of time samples,k0=1,. . .,K, whereK is the total number of frequency samples, andσ = {1, 2, 4,. . ., 2log2N −1}.
The proposed greedy search algorithm is an extension of the orthogonal matching pursuit (OMP) described in [59] to the time-frequency domain. The orthogonal matching pursuit adds a least-squares minimization to each step of MP to obtain the best approximation over the atoms that have already been chosen. This revision significantly improves the convergence speed of the algorithm. For a given time-frequency matrix, TFD, the search for logons that best describe the surface can be summarized as follows.
(1) Initialize the residue asR0=TFDand setl=0 and
D=D.
(2) At the lth iteration, find the Gabor logon over the whole overcomplete dictionary, that is, over all (n0,k0,σ), that has the largest inner product with the residue time-frequency surface,Rl
Gl(n,k)=argmax
Gi(n,k)∈D Rl,G
i
=argmax
Gi(n,k)∈D
N
n=1 K
k=1
Rl(n,k)Gi(n,k) .
(12)
(3) Compute the approximation at thelth step,Al, as
Al=argmin
A
TFD−A2, (13)
whereA ∈span{Gi,i =1, 2,. . .,l}. This problem is
solved using a least squares optimization approach.
(4) Subtract the approximation,Al, from the residue to
compute the new residue time-frequency distribu-tion at thel+ 1th iteration
Rl+1(n,k)=Rl(n,k)−Al(n,k). (14)
(5) Incrementlby 1, setD=D\Gl.
(6) Go back to step 2 until a predetermined number of atoms is selected or the normalized mean squared
error (NMSE) between the TFD and the approxi-mation at the lth iteration is below a predetermined threshold, that is,
NMSE=
TFD−Al2 2 TFD22
=
N n=1
K k=1
TFD(n,k)−Al(n,k)2
N n=1
K
k=1TFD2(n,k)
< γ.
(15)
NMSE is a measure of how close the approximation from the dictionary is to the original time-frequency distribution. Since the mean square error is normal-ized by the energy of the original TFD, it is always between 0 and 1.
4. Simulated and Biological Data Analysis
4.1. Description of Biological Data. The biological data used in this paper has been previously presented utilizing PCA on the time-frequency plane approach, and thus we will only detail the relevant parameters here. The reader is directed to the previously published paper for greater detail [41]. The sample consisted of twins in the Minnesota Twin Family Study (MTFS), a longitudinal and epidemiological investigation of the origins and development of substance use disorders and related psychopathology. All male and female twin participants for whom ERP data were available from the study’s psychophysiological assessment served as subjects for this investigation. This sample combined subjects from the two age cohorts of the MTFS. Subjects in one cohort were 17 years old at intake whereas subjects in the other were approximately 11 years old at intake. Data for this younger cohort came from a follow-up assessment conducted when subjects were approximately 17 years old. The sample thus comprised 2,068 17-year-old adolescents in all (mean age= 17.7; SD=0.5; range=16.7 to 20.0).
For each trial, 2 s of EEG, including a 500 ms prestimulus baseline, were collected at a sampling rate of 256 Hz. EEG data were recorded from three parietal scalp locations, one on the midline (Pz) and one over each hemisphere (P3 and P4). Consistent with the previous report, only data from the Pz electrode is reported here. Similarly, although ERPs to standard (frequent) stimuli were collected, they were not analyzed for the current paper; target condition responses serve as the basis for all decompositions and analyses presented. Therefore, the analysis in this paper focuses on data reduction for ERPs collected across multiple subjects from a single channel. However, the methods developed can easily be extended to single subject and/or multiple channel data.
Principal component decompositions were employed to evaluate the proposed approach. For the purposes of this study, decompositions for condition-averaged data were conducted on narrow time and frequency ranges, to focus on lower frequency delta and theta activity. Condition-averaged ERPs were constructed separately for easy and hard discrimination conditions. These included frequencies ranging from 0 to 5.75 Hz and time ranging from stimulus onset to 1000 ms poststimulus. The range was narrowed to focus on the time-frequency range containing the majority of variance: theta, delta, and low frequency activity.
4.2. Description of Simulated Data. Two simulated datasets were employed in the current paper. As with the biological data, these datasets were employed previously with PCA approach alone [37]. Briefly, the two sets included are 3-logons and 3-3-logons with noise. All simulated sets were 100 Hz sampled signals of 1000 ms, with the first and last 100 ms discarded after the TFD is computed to remove edge effects. The first simulated dataset contains 3 logons with clearly separated time and frequency centers: 30 Hz/100 ms, 20 Hz/400 ms, and 10 Hz/700 ms. For 3-logons with noise, noise was added at the 4 dB signal to noise level. In all simulations, each signal entered was assigned to a different simulated topographical region, to simulate the activity from different brain areas. To accomplish this, the signals were divided into 63 simulated channels creating a 7×9 grid within which differential weightings could be applied. Each signal entered, that is, each logon, was weighted by a 4×4 grid differentially located within the overall 7×9 grid. The differential loadings were implemented to simulate a signal with more focal activity that decays in topographic space. The simulated datasets each contained 7560 total waveforms, comprised of 120 trials by 63 electrodes.
4.3. Results. In this section, we will present the results of applying the PCA-Gabor method on both simulated data and ERP signals described above. In the PCA-Gabor analysis, we will focus on extracting the “best” logon fit to the principal component surface. Extracting the “best” logon offers a way of parameterizing the PCs using the time, frequency and scale parameters of the logon as well as serving as a denoising tool since the “best” logon will focus on representing the actual signal energy as opposed to background noise. Through
Table1: The comparison of the variance explained by the PCA, PCA-Gabor, and SMP-Gabor for two simulated datsets: 3-logons and 3-logons in noise.
Data Set PCA PCA-Gabor SMP-Gabor
3 Logons 99.97% 95.73% 95.73%
3 Logons in Noise 81.38% 90.78% 38.20%
this analysis, we will show the effectiveness of the PCA-Gabor method both as a modeling/data reduction tool and a denoising tool. The proposed method also offers an alternative to previous ERP studies that use matching pursuit to decompose each signal individually [13]. This analysis has the disadvantage of being computationally expensive and extracting a large number of logons to represent a collection of signals. Since a comparison between matching pursuit at the individual signal level and the PCA-Gabor method would not be helpful due the number of logons extracted being much larger for the MP, we compare the PCA-Gabor method to the simultaneous matching pursuit with Gabor dictionary (SMP-Gabor) and to previous results obtained by the PCA method on the time-frequency plane [37].
4.3.1. Analysis of Simulated Data. The different methods were first evaluated for simulated data made up of Gabor logons. For this analysis, decompositions from two simulated datasets containing 3-logons with and without additive white noise were used. The PCA approach involved selecting the three time-frequency PCs with the highest eigenvalues. The PCA-Gabor approach extracted the “best” Gabor logon for each of the three PCs yielding three logons. Finally, the SMP-Gabor method extracted the best three logons that explained the whole data set. For the 3-logons without noise all of the methods explained more than 95% variance of the data set with the PCA performing the best (Table 1). The logons extracted from PCA-Gabor and SMP-Gabor were identical (see Figure 1) explaining exactly the same amount of data variance indicating that under ideal conditions PCA-Gabor performs as well as the standard SMP-Gabor.
For 3-logons in 4 dB noise, similarly, three components were extracted by each of the algorithms, that is, three PCs with PCA, three logons fitted to the PCs with PCA-Gabor, and three logons fitted to the whole dataset with SMP-Gabor. The extracted components were evaluated in terms of the amount of signal variance they captured by projecting the components onto the original 3-logon dataset. FromTable 1, it can be seen that PCA-Gabor captures the most amount of signal variance with PCA coming in second. The logons extracted from the SMP-Gabor method can only explain 38.20% of the total signal variance since the algorithm focuses on extracting components that capture the most amount of common variance in the data (whereas PCA is covariance-based), which in this case corresponds to noise.
Grand average
No noise
(Hz)
0 20 40
(ms)
0 400 800
With noise
(Hz)
0 20 40
(ms)
0 400 800
TF amplitude
High
Low
(a) Decomposition
No noise With noise
PCA PCA-gabor SMP-gabor PCA PCA-gabor SMP-gabor
(Hz)
0 20 40
(Hz)
0 20 40
(Hz)
0 20 40
Com
p
on
en
ts
(ms)
0 4 8
×102
(ms)
0 4 8
×102
(ms)
0 4 8
×102
(ms)
0 4 8
×102
(ms)
0 4 8
×102
(ms)
0 4 8
×102 TF amplitude
High
Low
(b)
Figure1: A comparison of the principal components (PCA), Gabor logons extracted from the principal components (PCA-Gabor) and Gabor logons extracted by Simultaneous Matching Pursuit using a Gabor dictionary (SMP-Gabor) from two simulated datasets (3-logons with no noise and 3-logons with noise).
Table2: Partial eta-squared (η2
p) values from a repeated-measures
general linear model (GLM) for the three methods for ERP analysis. Multivariate tests indicate statistical values across all components.
Note:∗P < .05,∗∗P < .01, and∗P < .001.
Factors Sex RT Difficulty
Multivariate PCA .102∗∗∗ .105∗∗∗ .088∗∗∗
Multivariate PCA-Gabor .105∗∗∗ .109∗∗∗ .091∗∗∗
Multivariate SMP-Gabor .086∗∗∗ .104∗∗∗ .084∗∗∗
4.3.2. Analysis of Biological Data. For the biological data, first we will compare the variance characterized using the different approaches. We extract 11 PCs using PCA, the 11 logons extracted from these PCs using PCA-Gabor, and 11 logons that best explain the energy of the whole dataset using the SMP-Gabor method. Once the different components are extracted, they are projected onto each of the 8328 condition-averaged ERP waveforms. For the three methods compared in this paper, PCA, PCA-Gabor and SMP-Gabor, PCA explained most of the data variance with 91%. For SMP-Gabor, the variance explained was 81% whereas for PCA-Gabor it was 70%.
While the PCA explains the most overall variance in the data, and PCA-Gabor the least, additional analysis is needed to evaluate the methods in terms of experimentally
relevant variance. To accomplish this, the three methods were compared for statistical separation using three common variables: sex, reaction time, and task difficulty. Because the activity extracted from the three methods covers much of the same time-frequency range, it is expected that the three methods should provide similar statistical effects. Statistical evaluation was conducted using a repeated-measures general linear model (GLM) including sex, reaction time, and task difficulty. A separate GLM was conducted for the sets of 11 components from each method. The design was Sex (male/female) by RT (reaction time; continuous) by task Difficulty (easy/hard; the within-subjects repeated measure). These main effects were highly significant for all three methods, confirming that similar experimentally relevant activity was extracted. Partial eta-squared (η2
p) values for
the three methods are summarized in Table 2. Here, for sex, RT, and difficulty, the nominal order of the amount of experimentally relevant variance in the statistical effects was the same, largest in the PCA-Gabor, next in the PCA, and the least in the SMP-Gabor.
Table3: A qualitative comparison of the three data reduction methods discussed in this paper.
Method/Properties Data dependent Time-Frequency
Parametrization Computation time Variance versus Covariance
PCA Yes No 3.2 sec Covariance
PCA-Gabor Yes Yes 11.6 sec Covariance
SMP-Gabor No Yes 1276 sec Variance
Grand average
Mi
cr
ov
o
lt
s
0 10 20
0 200 400 600 800 1000
(Hz)
0 2 4
TF amplitude
0.3
0 (a)
Decomposition
PCA PCA-gabor SMP-gabor
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4 Component 1
Component 11
(ms)
0 1
×103
(ms)
0 1
×103
(ms)
0 1
×103 TF amplitude
0.3
0
(b)
Grand average
Mi
cr
ov
o
lt
s
0 10 20
0 200 400 600 800 1000
(Hz)
0 2 4
TF amplitude
0.3
0 (a)
Decomposition
MP-grand average
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4
(Hz) 02
4 Component 1
Component 11
(ms)
0 1
×103 TF amplitude
0.3
0
(b)
Figure3: 11 Gabor logons extracted from the grand average of ERP signals.
This is analogous to the results of the simulated data with noise, where the PCA-Gabor method extracted the most signal power in terms of experimentally relevant variance, while excluding the largest amount of noise power in terms of experimentally irrelevant variance. Thus, in these terms, the PCA-Gabor method was the most optimal among the three methods.
Finally, it is important to compare the different ap-proaches in terms of computational complexity. All of the
Table4: The parameters (time center (samples),n0, frequency center (Hz), f0, and scale parameterσ) of the 11 logons extracted by
PCA-Gabor from the 11 PCs, SMP-PCA-Gabor from the 8238 TFD surfaces, and MP-PCA-Gabor from the grand average of the 8238 time-frequency surfaces.
PCA-Gabor SMP-Gabor Grand average-Gabor
Logon number n0 f0 σ n0 f0 σ n0 f0 σ
1 6 4.3128 1 15 1.265 8 15 1.265 8
2 7 1.4376 4 12 1.9837 8 12 1.955 8
3 16 0.9584 8 19 0.7188 16 19 0.7188 16
4 11 2.8752 1 12 2.7312 2 12 2.875 8
5 13 3.5940 1 27 1.4375 8 29 1.2363 4
6 5 3.3544 1 10 1.61 4 9 1.4375 4
7 14 2.1564 2 8 2.53 2 6 4.1113 1
8 8 2.3960 2 15 2.7025 1 12 4.1113 1
9 27 1.4376 8 19 1.9837 2 14 0.3738 8
10 23 1.9168 2 6 3.7950 1 19 2.3288 2
11 19 1.4376 4 9 3.5938 1 3 1.9837 4
by a large margin is the SMP-Gabor method, which was computationally expensive due to the core search algorithm required.
Table 3 summarizes the key properties of the three methods compared in this section in terms of their data dependence, time and frequency parametrization, compu-tational efficiency for the ERP data set and whether the resulting decomposition is based on explaining the most variance or covariance in the data.
4.4. Discussions. Several overall trends in the results are important to detail. First, the PCA-Gabor characterized more experimental variance than the PCA, with less of the overall raw data variance. This suggests that the Gabor decom-position of the PCA represents the relevant information obtained in the PCA, supporting the view that the activity extracted by PCA largely contains activity that conforms to Gabor constraints. Second, because the PCA-Gabor explains nominally more experimentally relevant variance, and outperforms the SMP-Gabor, while using less of the raw data variance than either, it supports the contention that this approach produces a more optimal Gabor decomposition of the collection of signals than the standard matching pursuit. Finally, it is interesting to specifically consider the fact that the PCA-Gabor method explains the least amount of data variance compared to the other two methods. The components extracted by PCA explain most of the data variance since PCA is designed to maximize the variance explained and extracts components that are orthogonal to each other. The PCA-Gabor method, on the other hand, approximates the energy of each principal component with a single logon and thus, the total variance explained is lower than the original principal components. However, this method has the advantage of retaining the signal variance and getting rid of the noise variance, thus acting as an effective denoising method, while also parameterizing the time-frequency surfaces. The third method, SMP-Gabor, explains more of the data variance compared to PCA-Gabor but has less experimental condition sensitivity (e.g., statistical significance). This increased variance and reduced
sensitivity can be explained by looking at the Gabor logons extracted from the biological data by PCA-Gabor and SMP-Gabor methods shown inFigure 1. Although the two methods extract some common logons, SMP-Gabor method emphasizes the low frequency activity. The first three logons extracted by this method are low frequency logons with a large time spread. The major reason for this is that the SMP-Gabor method operates entirely on the variance, and thus focuses the most on the high-amplitude (i.e., variance) low-frequency area of the surface. The PCA approaches, on the other hand, operate on covariance, which focuses more on activity that is functionally related (i.e., covaries). This point is also supported by the Gabor decomposition of the grand average of the 8328 waveforms given in
Figure 3.Table 4compares the parameters of the 11 logons extracted from the 11 PCs using the PCA-Gabor method, from the 8238 TFD surfaces using the SMP-Gabor method and from the grand average of the 8238 waveforms using standard MP-Gabor. This table indicates that there are some commonalities between the SMP-Gabor and MP-Gabor on the grand average surface since they extract similar logons describing the low frequency activity, for example, logons 1-3 are almost identical.
5. Conclusions
the principal components instead of the actual data helps to extract parameters that represent the covariation among observations rather than characterize the average energy across observations. This property of PCA-Gabor becomes especially important when there is considerable noise in the data since standard matching pursuit algorithms will focus on fitting parameters to capture the most amount of energy, which in this case may be noise components. This phenomenon exhibited itself in the analysis of the biological data as PCA-Gabor most effectively differentiated between the experimental conditions with the least amount of data variance, or in other words capturing the least amount of noise, compared to the other two methods. This was mainly because the extracted logons explained the main effects described by the principal components with higher signal-to-noise ratio (most experimentally relevant variance).
Future work will focus on the extensions of the proposed methods to different data factorization approaches such as the ICA. For ERP data collected over multiple channels, spatial ICA may be used as an alternative to PCA and the proposed data reduction method can be applied onto the independent components. Future work will also evaluate the Gabor parameters in relation to well-known cognitive ERP events such as P300, as well as ERP events with known specific neurological origins, such as anterior cingulate cortex activation as measured in the error-related negativity (ERN) paradigm.
Acknowledgments
This work was in part supported by Grants from the National Science Foundation under CAREER CCF-0746971, National Institutes of Health NIDA13240, NIDA05147, NIDA024417, NIAA09367, and K08MH080239.
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